当前位置: X-MOL 学术J. Stat. Phys. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
The Asymptotics of the Clustering Transition for Random Constraint Satisfaction Problems
Journal of Statistical Physics ( IF 1.3 ) Pub Date : 2020-10-24 , DOI: 10.1007/s10955-020-02635-8
Louise Budzynski , Guilhem Semerjian

Random Constraint Satisfaction Problems exhibit several phase transitions when their density of constraints is varied. One of these threshold phenomena, known as the clustering or dynamic transition, corresponds to a transition for an information theoretic problem called tree reconstruction. In this article we study this threshold for two CSPs, namely the bicoloring of $k$-uniform hypergraphs with a density $\alpha$ of constraints, and the $q$-coloring of random graphs with average degree $c$. We show that in the large $k,q$ limit the clustering transition occurs for $\alpha = \frac{2^{k-1}}{k} (\ln k + \ln \ln k + \gamma_{\rm d} + o(1))$, $c= q (\ln q + \ln \ln q + \gamma_{\rm d}+ o(1))$, where $\gamma_{\rm d}$ is the same constant for both models. We characterize $\gamma_{\rm d}$ via a functional equation, solve the latter numerically to estimate $\gamma_{\rm d} \approx 0.871$, and obtain an analytic lowerbound $\gamma_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$. Our analysis unveils a subtle interplay of the clustering transition with the rigidity (naive reconstruction) threshold that occurs on the same asymptotic scale at $\gamma_{\rm r}=1$.

中文翻译:

随机约束满足问题聚类转移的渐近性

随机约束满足问题在约束密度变化时表现出几种相变。这些阈值现象之一,称为聚类或动态转换,对应于称为树重建的信息论问题的转换。在本文中,我们研究了两个 CSP 的阈值,即具有约束密度 $\alpha$ 的 $k$-均匀超图的双色,以及平均度为 $c$ 的随机图的 $q$-着色。我们表明,在较大的 $k,q$ 限制中,聚类转换发生在 $\alpha = \frac{2^{k-1}}{k} (\ln k + \ln \ln k + \gamma_{\ rm d} + o(1))$, $c= q (\ln q + \ln \ln q + \gamma_{\rm d}+ o(1))$,其中 $\gamma_{\rm d} $ 是两个模型的相同常数。我们通过函数方程表征 $\gamma_{\rm d}$,对后者进行数值求解以估计 $\gamma_{\rm d} \approx 0.871$,并获得解析下界 $\gamma_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \约 0.812$。我们的分析揭示了聚类转换与刚性(朴素重建)阈值之间的微妙相互作用,该阈值发生在 $\gamma_{\rm r}=1$ 的相同渐近尺度上。
更新日期:2020-10-24
down
wechat
bug